1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Characteristic polynomial help

  1. Nov 1, 2005 #1

    Note: There is exactly one real zero of the characteristic polynomial and it
    has multiplicity 3 (it is a positive integer!). The other zeros are complex
    and they have multiplicity 2.

    Sadly I missed this lecture day, and am unsure of where to start. Any diffeq demi-gods out there?
  2. jcsd
  3. Nov 1, 2005 #2


    User Avatar
    Homework Helper

    For an equation of order n, if a root (say r1) has a multiplicity s (s =< n), where x is the independent variable

    [tex] e^{r_{1}x}, xe^{r_{1}x}, x^{2} e^{r_{1}x}, ..., x^{s-1} e^{r_{1}x} [/tex]

    For complex roots, let's say [itex] a+bi [/itex] is repeated s times, then the complex conjugate [itex] a-bi [/itex] is also repeated s times, therefore the solutions for real valued functions, where x is the independent variable:

    [tex] e^{ax} \cos{bx}, e^{ax} \sin{bx}, xe^{ax} \cos{bx}, xe^{ax} \sin{bx},..., x^{s-1} e^{ax} \cos{bx}, x^{s-1} e^{ax} \sin{bx} [/tex]
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?